$A$ straight line through the point $(1, 1)$ meets the $x$-axis at $A$ and the $y$-axis at $B$. The locus of the mid-point of $AB$ is

If the line segment joining the points $(1,0)$ and $(0,1)$ subtends an angle of $45^{\circ}$ at a variable point $P$,then the equation of the locus of $P$ is

The locus of a point $P(x, y)$ which moves in such a way that the segment $OP$,where $O$ is the origin $(0, 0)$,has a slope of $\sqrt{3}$ is:

$A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P(x, y)$ onto the lines $AB, BC$ and $CA$ respectively. If $a, b, c$ are in arithmetic progression,then the locus of $P$ is

$A$ variable line passes through a fixed point $(x_{1}, y_{1})$ and meets the axes at $A$ and $B$. If the rectangle $OAPB$ is completed,the locus of $P$ is,($O$ being the origin of the system of axes).

$A \equiv (\cos \theta, \sin \theta)$ and $B \equiv (\sin \theta, -\cos \theta)$ are two points. The locus of the centroid of $\triangle OAB$,where $O$ is the origin,is